On the Absence of Identifiable Manifolds in Finite-Max Composite Optimization

Minima Selection in Stochastic Optimization: A Long-Time, Small-Stepsize Perspective

Riemannian ascent--descent for nonconvex nonconcave minimax landscapes: convergence to basin saddle points and applications to distributionally robust optimization

We study a class of distributionally robust optimization (DRO) problems for the statistical risk problem, formulated as minimax problems over the product of a Euclidean space and a Riemannian manifold. Because the resulting minimax landscape is nonconvex nonconcave in general, no globally convergent first order method is known to be available. We instead introduce the notion of a \emph{basin saddle point}, a Nash equilibrium defined locally on the Cartesian product of a $\delta$ basin around a connected component of the local minima critical set and a geodesic ball on the measure manifold. We develop an abstract convergence framework for a Riemannian gradient ascent multistep descent iteration to a basin saddle point under a local \L{}ojasiewicz type growth condition, with exponent $\beta \in (1,2]$, in the $\delta$ basin around connected components of the local minima critical sets. Under Lipschitz regularity of critical sets we establish linear convergence for $\beta = 2$ and polynomial convergence for $\beta \in (1,2)$ to a basin saddle point, with explicit dependence on the sectional curvature of the manifold. We then instantiate this framework for the statistical risk DRO problem over Gaussian measures, where the ambiguity set is naturally modeled as the product of Euclidean space and the Bures Wasserstein manifold of covariance matrices, which we relax to a penalized DRO formulation. We derive nonasymptotic Hessian estimates for the resulting Lagrangian, establish existence and local uniqueness of its maximizer, and prove that an alternating Riemannian gradient scheme converges to a basin saddle point of the penalized DRO problem, recovering the linear and polynomial rates of the abstract theory with all constants explicit in terms of data dimension, loss moments, and the reference covariance.

Maximizing Families and Typical Periodic Optimization for Almost Additive Sequences

In this paper, we study typical periodic optimization (TPO) for almost additive potentials in two perturbation spaces. Our main setting is the Banach quotient $\mathcal E_{\rm orb}(X,T)$ of orbit-Lipschitz almost additive potentials, where we extend the theory of maximizable sets and countable maximizable families developed by W. Huang, O. Jenkinson, L. Xu and Y. Zhang [Typical periodic optimization for dynamical systems: symbolic dynamics, Invent. Math. 245 (2026), 1--63], and establish a global structural theorem. For a countable maximizable family, global TPO holds if every non boundary member has $X$-extendable TPO and the boundary region has empty interior. As an application, we construct a compact system with global TPO for which $\mathcal E_{\rm orb}(X,T)$ is infinite-dimensional and the maximizing periods in open locking regions are unbounded. For a fixed almost additive potential $\Phi$, we also develop relative TPO theory on its Lipschitz leaf. When $\Phi=0$, this framework reduces to classical Lipschitz TPO. We prove the corresponding leafwise structural theorem and give a non additive rank-one matrix example on a full shift.

Infinitely many graph manifolds with unique geometrical piece that admit arbitrarily many Anosov flows

Anosov flows have a long and rich history, firstly motivated by the study of geodesic flows in negative curvature surface by Anosov and Sinai. Not every closed manifold admits an Anosov flow for well-known reasons: the fundamental group of a 3-manifold \(M\) admitting an Anosov flow must have exponential growth, and \(M\) must be universally covered by \(\mathbb{R}^{3}\). Nevertheless, there are sufficient mechanisms for constructing distinct Anosov flows on admissible 3-manifolds, such as Dehn-Goodman-Fried surgery or playing with hyperbolic building blocks. A central problem in the field has been to determine the number of Anosov flows that can be supported by a single manifold. The question of whether there exists an infinite set of pairwise non-equivalent Anosov flows on a 3-manifold remains open to this day. However, there are several papers proving the existence of a manifold $M_n$ that admits $n$ pairwise inequivalent Anosov flows for any natural number $n$. In all known examples, the manifolds $M_n$ are composed of several geometric pieces. In the present paper, we prove the existence of a countable number of graph manifolds $M_{k,n}$, $k \in \mathbb{N}$ with a single geometric piece, each of which admits $n$ pairwise non-equivalent transitive Anosov flows. All previously known constructions of different flows on the same graph manifold were based on gluing geodesic flows. The nature of the flows constructed in this paper is completely different; they are constructed from a single hyperbolic plug, which is a suspension over a Morse-Smale diffeomorphism on a surface.

Robustness Analysis via Horofunction Compactification

Robustness analysis plays a central role in the verification and design of computational and hybrid systems, particularly when system behaviour depends continuously on parameters subject to perturbation. Existing domain-theoretic frameworks provide a principled foundation for reasoning about such perturbations via monotone maps on lattices of closed sets. However, these frameworks face significant limitations when the underlying state space is not locally compact, as is the case for the infinite-dimensional spaces that arise in analysis, machine learning, and control theory (e.g., $\ell_p$ and $L_p$ spaces). In these settings, the lattice of closed subsets fails to be continuous, and classical compactifications either sacrifice precision or lack computable structure. We propose Gromov's horofunction compactification as a new tool for robustness analysis over a class of separable metric spaces of practical importance, including separable reflexive Banach spaces. Given a metric space $\mathbb{S}$, we show that its horofunction extension yields a compact metric space together with a Lipschitz embedding, which enables robust approximations of monotone maps via Scott-continuous maps on the compactified domain. For separable spaces, the horofunction compactification is metrizable, which provides a path toward effective domain-theoretic constructions.

First-Order Optimization under Uniform Nondegeneracy: Geometry, Computation, and Information

Strongly convex minimization and its natural indefinite extension to strongly convex--strongly concave minimax problems combine quantitative control of curvature with a prescribed curvature orientation. We disentangle these two roles by retaining uniform nondegeneracy alone: curvature remains uniformly separated from zero but may have either sign, with no prescribed positive--negative splitting. Surprisingly, a large part of the familiar theory nevertheless re-emerges. We first derive an intrinsic formulation through first-order secant inequalities, making the gradient on $\mathbb{R}^d$ a global bi-Lipschitz homeomorphism and yielding a unique stationary point. We then pair signed Moreau envelopes to construct a smooth scalar merit that recovers the missing descent geometry at both zeroth and first order, and the resulting paired proximal descent method achieves global linear convergence with dimension-free first-order oracle complexity. Meanwhile, we show that this tractability can break down on restricted domains: merely assuming the existence of a stationary point in the domain may lead to the curse of dimensionality, even with access to an infinite-order oracle. To overcome this information barrier, we introduce certified feasibility, an observable localization condition that enables feasible continuation. Together, these results establish a first-order optimization theory under uniform nondegeneracy that spans geometry, computation, and information.